Cellular automata, percolation and dynamical dichotomies
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866912118187491328 |
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| author | Barbieri, Sebastián García-Ramos, Felipe Taati, Siamak |
| author_facet | Barbieri, Sebastián García-Ramos, Felipe Taati, Siamak |
| contents | We establish a connection between percolation on the Cayley graphs of a group and the dynamical diversity of cellular automata on that group. Specifically, we demonstrate that Gilman's dichotomy between equicontinuity and sensitivity with respect to Bernoulli measures holds on a finitely generated group if and only if the group has a trivial percolation threshold. Consequently, we show that a countable group satisfies Gilman's dichotomy if and only if it is locally virtually cyclic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_23770 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cellular automata, percolation and dynamical dichotomies Barbieri, Sebastián García-Ramos, Felipe Taati, Siamak Dynamical Systems Group Theory Probability 37B15, 37A50, 60K35 We establish a connection between percolation on the Cayley graphs of a group and the dynamical diversity of cellular automata on that group. Specifically, we demonstrate that Gilman's dichotomy between equicontinuity and sensitivity with respect to Bernoulli measures holds on a finitely generated group if and only if the group has a trivial percolation threshold. Consequently, we show that a countable group satisfies Gilman's dichotomy if and only if it is locally virtually cyclic. |
| title | Cellular automata, percolation and dynamical dichotomies |
| topic | Dynamical Systems Group Theory Probability 37B15, 37A50, 60K35 |
| url | https://arxiv.org/abs/2410.23770 |